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An asymptotic-induced scheme for nonstationary transport equations with thediffusion scaling is developed. The scheme works uniformly for all ranges ofmean free paths. It is based on the asymptotic analysis of the diffusion limit ofthe transport equation. A theoretical investigation of the behaviour of thescheme in the diffusion limit is given and an approximation property is proven.Moreover, numerical results for different physical situations are shown and atheuniform convergence of the scheme is established numerically.

An asymptotic-induced scheme for kinetic semiconductor equations with the diffusion scaling is developed. The scheme is based on the asymptotic analysis of the kinetic semiconductor equation. It works uniformly for all ranges of mean free paths. The velocity discretization is done using quadrature points equivalent to a moment expansion method. Numerical results for different physical situations are presented.